The Mathematics of Financial Modelingand Investment Management

4-PrincipCalculus Page 137 Friday, March 12, 2004 12:39 PM

Principles of Calculus 137

∞

hx()= (f∗g)()x = ∫fx t ( – )gt()dt

• ∞

It can be demonstrated that the following property holds:

L[hx()] = L[f∗g]= L[fx()]L[gx()]

As we will see in Chapter 9, when we cover differential equations,

these properties are useful in solving differential equations, turning the

latter into algebraic equations. These properties are also used in repre-

senting probability distributions of sums of variables.

Fourier Transforms

Fourier transforms are similar in many respects to Laplace transforms.

Given a function f, its Fourier transform fˆ (ω) = F[f(x)] is defined as the

integral

fˆ ω ()]=

+∞

()= F[fx ∫ e –^2 πiωxfx

• ∞

()dx

if the improper integral exists, where iis the imaginary unity. The Fou-

rier transform of a real-valued function is thus a complex-valued func-

tion. For a large class of functions the Fourier transform exists and is

unique, so that the original function, f, can be recovered from its trans-

form, fˆ.

The following conditions are sufficient but not necessary for a func-

tion to have a forward and inverse Fourier transform:

∞

■ ∫ fx()dxexists.

• ∞
  ■ The function f(x) is piecewise continuous.
  ■ The function f(x) has bounded variation.

The inverse Fourier transform can be represented as:

∞

fx ()] = e

2 πiωx

()= F fˆ ω

• 1

[fˆ ω ∫ ()dω

• ∞